Fractals in Physics

In a famous book in 1975, B. Mandelbrot introduced fractals into our worldview. The spread of this concept has followed paths as strange as the objects themselves. Starting out as a mathematical notion, the concept slowly spread to the various branches of science. Not that these objects were not soon recognized in physics or biology, but a metaphysical connotation made the idea suspect in the eyes of many scientists. It's a story that has already unfolded in other circumstances, and for other objects. So long ago, in fact, that we've almost forgotten it: the Greek sceptics denied the usefulness of Euclidean Geometry, because this science was based, in their view, on abstract and unimaginable concepts. "The straight line is inconceivable", wrote Sextus Empiricus, arguing that such an infinite object of zero thickness could not be represented – or even mentally –. With fractals, we're faced with similar vertigo. Here we find a new kind of infinity, which was quickly reclaimed by our collective unconscious; I'm not even going to mention the introduction of fractals into art, which has brought this notion into even sharper focus with the general public. It's easy to see how much effort a scientist has to make to free himself from such a metaphysical burden and remain at a pragmatic level. And we'll forgive those who were once tempted to say with a certain disdain: "People now see fractals everywhere! Even if this kind of thinking has slowed down the spread of an idea that is nonetheless proving more fruitful by the day.

So, should we see fractals everywhere, or should we deny their real existence? Fortunately, there is a "middle way": real fractals exist within a certain range of lengths. Within these limits, we see a fractal object, and the physical properties faithfully reflect the fractality of the structure. Beyond that, the object becomes common again. It's this approach, resolutely oriented towards physics, that we're going to look at in this article, on real examples, and the reader will therefore be spared this trying mental exercise: trying to imagine these objects which, like the straight line, must in principle be material and structured, although of exactly zero volume...

The deliberate bias of this article is therefore limited to the volumetric fractal objects studied in physics. The term "aggregates" will be used, in the general sense of objects made up of microscopic entities (particles). This means that, for reasons of consistency of writing, the description of surfaces and fractal lines will be excluded from this study, even though they do, of course, in principle have a place in physics. It's important to realize that there are entire books devoted to the simple geometry of fractals and that, in order to go into a little more detail, we're obliged...